L(s) = 1 | − 5-s + 3·7-s − 11-s − 4·19-s + 25-s + 6·29-s − 7·31-s − 3·35-s + 2·37-s + 2·41-s + 43-s − 10·47-s + 2·49-s + 6·53-s + 55-s + 6·59-s − 3·61-s − 7·67-s + 4·71-s + 15·73-s − 3·77-s − 5·79-s − 6·83-s + 10·89-s + 4·95-s − 19·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.301·11-s − 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.25·31-s − 0.507·35-s + 0.328·37-s + 0.312·41-s + 0.152·43-s − 1.45·47-s + 2/7·49-s + 0.824·53-s + 0.134·55-s + 0.781·59-s − 0.384·61-s − 0.855·67-s + 0.474·71-s + 1.75·73-s − 0.341·77-s − 0.562·79-s − 0.658·83-s + 1.05·89-s + 0.410·95-s − 1.92·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73006530954735, −12.37108208154993, −11.85565723977778, −11.42714614243169, −10.94361056983941, −10.76336000972501, −10.13554815505607, −9.675807349534689, −9.044967430009166, −8.580540271670595, −8.218787437607193, −7.835842356978875, −7.364361293278399, −6.782471603804403, −6.387980010695007, −5.683025720797776, −5.252822202107569, −4.717492154844924, −4.358189708887614, −3.805015997571344, −3.221526994588276, −2.542443977974905, −2.033822338199802, −1.463485689882956, −0.7646784629936721, 0,
0.7646784629936721, 1.463485689882956, 2.033822338199802, 2.542443977974905, 3.221526994588276, 3.805015997571344, 4.358189708887614, 4.717492154844924, 5.252822202107569, 5.683025720797776, 6.387980010695007, 6.782471603804403, 7.364361293278399, 7.835842356978875, 8.218787437607193, 8.580540271670595, 9.044967430009166, 9.675807349534689, 10.13554815505607, 10.76336000972501, 10.94361056983941, 11.42714614243169, 11.85565723977778, 12.37108208154993, 12.73006530954735